will it cause tranmission problem if towing a 5x10 trailer behind a ford Suv type vehicle?

would depend on the trailer weight and which SUV

The size of the trailer is immaterial. 

It is the WEIGHT and TONGUE WEIGHT of a trailer that is important.

Depends on four things - the weight of the towed trailer, the towing capacity of the vehicle, the roads it will be towed on (mountainous, hilly or flat) and the ambient outside temperature. Epic fail for not providing sufficient information dipshit. . 

Depends what you put on the trailer.  5x10 would take a very small car, say 1000kg. Every vehicle has a maximum towing weight , keep under that and it’ll be fine.

towing just the trailer no won't harm it 

If its rated to tow that, then it can tow it. It might take a few miles off the overall transmission life but nothing significant.

might need a bigger transmission cooler .......................

I don't see why it would . Doesn't sound like a big trailer .